Tolerance factor, deformation and Glazer notation

The highest symmetric ABO3perovskite crystal structure is cubic. Decreasing the size of the A cation induces a tilting of the BO6 octahedra to fill the extra interstices and reduce the unit cell volume inducing a distortion of the cubic structure. The metal-insulator temperatures TM I and the magnetic transition Neel temperature TN of nickelates, depend on this deformation which is quantified by the Goldschmidt tolerance factort defined:

t= (rRE+rO)

√2(rN i+rO) (2.1)

wherer_RE,r_{N i} andr_O are the ionic radii of the rare earth, the nickel and the oxygen atoms, respectively. The tolerance factort, Ni-O-Niγ angle, as well as metal-insulator transition temperature TM I and Neel temperature TN [47] for nickelates from R=lanthanum (La) to Lutetium (Lu) including the solid solution Sm1/2Nd1/2NiO3are summarized in table2.1and figure2.1.

Table2.1shows that a decrease of the tolerance factortmeans a reduction of the Ni-O-Niγangle, an increase of the deformation of the structure and a stronger stability of the insulating phase (Fig.2.1).

Figure 2.1: Phase diagram of Rare-Earth nickelates RNiO3. Tolerance factor (t), which indicates the deformation of the structure, as a function of temperature T.

At high temperatures, nickelates are paramagnetic metals indicated by red shading with rhombohedralR¯3ccrystal symmetry for R=La and orthorhombicP bnmcrystal symmetry for the rest of the familly. For R=Sm to Lu, the nickelates have a paramag-netic insulator phase indicated by yellow shading and characterized by a lowering of the symmetry toP21/c. At low temperatures, nickelates become antiferromagnetic, illustrated by blue shading. Here, theoretical prediction together with our Raman measurements suggest a lowering of the crystal symmetry toP2a21orP2acspace groups.

The phase diagram can be separated into four main parts:

• The ideal cubic perovskite structure, which would correspond to a Ni-O-Ni angle of 180^◦ has t=1.

• The least distorted member of the rare earth nickelates family is LaNiO3. The lanthanum (La) rare earth nickelate gives t=0.94, has γ=165.2^◦, and has a rhombohedral crystal structureR¯3cspace group.

It remains a paramagnetic metal even at low temperature. Indeed, the deformation is insufficient to induce an insulating phase due to the strong overlap of the 3dnickel and 2poxygen orbitals.

• Praseodymium (Pr) and neodymium (Nd) rare earth nickelates have t=0.925 (γ=158.7^◦) andt=0.915 (γ=157.2^◦), respectively. PrNiO3and NdNiO3 are both paramagnetic metals at high temperature with an orthorhombic crystal symmetry (Pbnm) and become antiferromagnetic insulators below 131 K and 202 K, respectively. Note that the metal-insulator temperature TM I and the Neel temperature TN coincide for

both compounds. It has been reported that the crystal structure adopts a monoclinicP21/c(orP21/nwhich is an alternative setting of the same structure and is closer to Pbnm) distortion in the insulating phase [37–39,41, 86]. However, has will be shown in chapter4our Raman measurements rather suggest a lowering of the crystal symmetry to P2a21orP2acspace groups, in agreement with group theory arguments.

• All other members of the family from samarium (Sm) to the smallest rare earth lutetium (Lu) lie in the range 0.894 (γ=152.6^◦)<t<0.851 (γ=143.4^◦). They are paramagnetic metals at high temperature and share the orthorhombicP bnmstructure of PrNiO3and NdNiO3. They undergo a metal-insulator transition below TM I with a monoclinic P21/cdistortion and remain paramagnetic until the antiferromagntic transition at the Neel temperature TN<TM I.

External or internal perturbations such as epitaxial strain, pressure, or rare earth substitution cause deformation of the structure and affect the electronic or magnetic properties of nickelates. As we mentioned in the last section, epitaxial strain has a dramatic impact on the electronic properties of nickelates. Many studies report a decrease of the metal-insulator transition temperature in NdNiO3 thin films compared to the value observed in bulk measurement on powders and single crystals, for both compressive (negative) and tensile (positive) strain. Some reports even demonstrate a vanishing of the low temperature insulating phase for compressive strain with NdNiO3 grown on YALO3 (-3.1%), on NdAlO3(-1.3%) and even on LaAlO3 (-0.5%) [55, 57, 58, 63, 64, 98]. The c axis parameter increases while theaandb axis parameters as well as the Ni-O-Niγ rotation angle decrease from tensile to compressive strains [58,61,96]. This competition between Ni-O bonds lengths and octahedral rotation leads to a complex and non-intuitive hybridization of orbitals [58, 96]. These results suggest that compressive strain promotes orbital overlap and broadens thed-band thereby favouring metallicity. Note also that tensile strain broadens the metal insulator transition in NdNiO3 [57,58,63]. Similar behaviour was observed for SmNiO3thin films [67], but it is not the case for all the nickelates. Indeed, in our own experiments which will be presented in the chapter3, we observe the opposite trend for LaNiO3 thin films. Our experimental observations are further supported by DFT calculations. Fowlie [6] argues in her thesis that this metallic behaviour, even at low temperature for compressively strained NdNiO3 thin films, can be caused by oxygen off-stoichiometry.

Indeed, nickelates are highly sensitive to the oxygen concentration and their resistivity can be tuned in this manner [69,72,74,97]. Dimensionality also has a important impact on the electronic properties and several studies have shown that reducing the thickness of nickelate films favors the insulating phase [6,63,65]. In the case of LaNiO3, Fowlie et al. show that the evolution

Nickelates Tolerance

PrNiO3 0.925 158.7 136.7 136.7

NdNiO3 0.915 157.0 203.0 203.0

Sm1/2Nd1/2NiO3 0.905 153.8 321.0 240.8

SmNiO3 0.894 152.6 403.0 230.8

EuNiO3 0.887 151.6 482.1 220.8

GdNiO3 0.881 150.6 511.6 185.4

DyNiO3 0.868 147.8 566.5 154.1

HoNiO3 0.866 147.4 574.0 148.4

ErNiO3 0.864 147.3 583.3 145.5

YNiO3 0.862 146.7 584.7 144.1

LuNiO3 0.851 143.4 600.1 131.2

Table 2.1: Tolerance factor t, Ni-O-Ni in-planeγangle, metal-insulator transition temperatureTM I (K) and Neel temperatureTN(K) for rare earth nickelates with R

= La, Pr, Nd,Sm_1/2N d_1/2, Sm, Eu, Gd, Dy, Ho, Er, Y, Lu. [47]

of the conductivity is due to three distinct types of local structure [68].

There are several propositions for the origin of the stabilization of the insulating phase in thin films. On one hand it may be caused by surface polar distortions coupled with octahedral rotations, which in turn decrease the Ni-O-Niγ angle. On the other hand, dimensional crossover may play a role [65, 99,100]. In both situations, the reduction of the bandwidthW induces the increase of TM I.

Applying pressure is another way to induce structural and therefore electronic change in these materials. Unlike biaxial strain which is strongly anisotropic, hydrostatic pressure induces a uniform stress along the three crystallographic directions. For example, several studies show a pressure-induced metal-insulator transition at room temperature on powders of SmNiO3, EuNiO3 and YNiO3 at about 2.5, 6 and 14 GPa, respectively [92, 101–106]. The metallic behaviour is explained by a gradual increase of the electronic bandwidth with increasing pressure and is accompanied by the P21/c to P bnm structural transition. For PrNiO3 and NdNiO3, a decrease of the metal-insulator temperature T_{M I} and even a complete suppression of the metal-insulator transition (TM I → 0) were observed at high pressure [91, 107–110]. It was also observed that pressure induces orthorhombic-rhombohedral phase coexistence followed by a pure rhom-bohedralR¯3c phase for pressures above 4 GPa for PrNiO3 [107, 111] and

NdNiO3[112] and above 30 GPa for SmNiO3 [105].

Finally rare earth substitution also permits tuning of the metal insu-lator transition by inducing structural deformations. In her thesis, Fowlie presents DC transport and Raman spectroscopy studies [6] on thin films of the mixture Nd1−xLaxNiO3which reveal the decrease of the metal-insulator temperature TM I with the increase ofxand the complete suppression of the metal-insulator transition for x > 0.2. She explains that a structural crossover between the P nma (orthorombic for x=0, NdNiO3) and R¯3c (rhombohedral forx=1, LaNiO3) space groups takes place. More precisely, betweenP nmaandC2/cbecause LaNiO3 adopts a monoclinic deformation withC2/c space group under biaxial strain, which is a subgroup of R¯3c. Raman measurements at room temperature do not show any phase coexis-tence, implying total samples homogeneity without any phase segregation and a rather gradual parameter evolution from the orthorombic to the monoclinic structure as a function of the x composition parameter. She argues that a structural transition occurs for x= 0.35 and can be driven by the allowed intermediate orthorhombic structure of space groupImma. She finally supposes that all compositions with x > 0.2 are blocked in a metastableC2/cstructure and bond disproportionation cannot occur, and thus it remains metallic.

To describe the deformation of the lattice and more precisely the ro-tation and tilt of the octahedra, it is useful to adopt the Glazer noro-tation.

The octahedra inclination can be decomposed by their rotation around the three crystallographic axisa,bandcwith the corresponding rotationsα,β andγ respectively (Fig.2.2a). A letter is repeated if the rotations are equal in magnitude for more than one axis. The octahedra rotate either in-phase (octahedra rotate in the same direction from layer to layer along the given rotation axis a, b orc) which is marked by a ” + ” sign exponent on the top of the letter, or out-of-phase (octahedra rotate in opposite direction from layer to layer along the given rotation axis) which is marked by a ”−” sign exponent. If there is no rotation the exponent is ”0”. The simplest example is illustrated by the octahedra in figure 2.2bwhich do not undergo any rotation (a⁰a⁰a⁰). While an octahedron on the top layer of figure2.2c rotates around the three crystallographic axis with the same magnitude, the corner sharing octahedron on the bottom layer rotates in opposite directions arounda,b andc. The Glazer notation is thereforea⁻a⁻a⁻. In figure2.2d, the top layer octahedron rotates around the three crystallographic axis, however, while the corner sharing octahedron on bottom layer rotates in opposite direction around aandb, it rotates in phase (and with different magnitude) alongc. The Glazer notation isa⁻a⁻c⁺.

Of course, the rotation of one octahedron induces opposite rotation of the four adjacent octahedra in the same layer.

Figure 2.2: Schematic view of the octahedral rotation.a, Characteristic octahedron and illustration of the tilt axis. b, Illustration of a⁰a⁰a⁰ Glazer notation. The octahedra undergo no rotation.c, Illustration ofa⁻a⁻a⁻Glazer notation. While a top layer octahedron rotates with the same magnitude around the three crystallographic axis, the corner sharing bottom layer octahedron rotates in opposite directions around the three axes.d, Illustration ofa⁻a⁻c⁺ Glazer notation. While a top layer octahedron rotates around the three crystallographic axis (aroundaandbwith the same magnitude), the corner sharing bottom layer octahedron rotates in opposite directions aroundaandband in phase aroundc.